Kathy has $1.00 in change. She has at least one of each coin less than half a dollar but does not have half a dollar. What is the least number of coins she has?

Question

ganeshie8 · Answer

what coins are available in US currency that are less than half dollar ?

anonymous · Answer

Nickel pennies dimes and quarters @ganeshie8

ganeshie8 · Answer

may i know how much they each value ?

ganeshie8 · Answer

1 dollar  = how many nickels
1 dollar = how many pennies

etc...

ganeshie8 · Answer

once we have that information, we can setup an equation and solve

anonymous · Answer

1 dollar= 20 nickels
1 dollar = 10 dimes
1 dollar= 100 pennies
1 dollar = 4 pennies

anonymous · Answer

I mean 4 quarters*

ganeshie8 · Answer

lets use this :) http://www.immihelp.com/newcomer/usa-currency-coins.html

ganeshie8 · Answer

convert everything into pennies, we can setup below linear diophantine equation :

$$\large  1p + 5n + 10d + 25q = 100$$

ganeshie8 · Answer

see if that looks fine ^^

anonymous · Answer

I'm a confused :/

ganeshie8 · Answer

since each nickel is worth 5 pennies, having $n$ nickels is same as having $5n$ pennies, right ?

ganeshie8 · Answer

similarly, having $d$ dimes is same as having $10d$ pennies,
and having $q$ quarters is same as having $25q$  pennies

anonymous · Answer

Yes I get it now :)

ganeshie8 · Answer

good :)  lets see if we can figure out something from the equation :

$$\large  1p + 5n + 10d + 25q = 100 $$

ganeshie8 · Answer

btw, what are we trying to find ?

ganeshie8 · Answer

this  :  ` What is the least number of coins she has?`

anonymous · Answer

Correct

ganeshie8 · Answer

Clearly she cannot have  `4` quarters (why ?)

anonymous · Answer

@ganeshie8 u r a pro sir

ganeshie8 · Answer

so hope that she has 3 quarters,
and lets find the possible dimes/nickels/pennies

anonymous · Answer

She has at least one of each coin

ganeshie8 · Answer

exactly ! so the maximum possible number of quarters is 3 only

anonymous · Answer

1 dime
2 nickels 5 pennies

ganeshie8 · Answer

that simplifies the equation to :
$$\large  1p +  5n + 10d = 25$$

ganeshie8 · Answer

again, she cannot have 2 dimes for the same reason, so the equation simplifies to :

q = 3
d = 1

$$\large  1p +  5n = 15$$

ganeshie8 · Answer

You're right !
3 quarters, 1 dime, 2 nickels, 5 pennies is the solution